The paper deals with price rivalry between two sellers of a differentiated good. Each firm occupies one of the two possible locations in the characteristics space considered. This resembles real situations like the choice between dichotomous substitutes, or examples coming from spatial economics. The consumers' population is partitioned in two groups according to preferences and the population density is constituted by two atoms. Transportation costs vary across consumers. Nevertheless the resulting firms' demand functions share the continuity property. At a non-cooperative equilibrium the prices of the two goods differ and some consumers who rank highest the more popular brand find it convenient to purchase the less expensive alternative. Higher one-firm concentration ratios are found to be consistent with lower equilibrium prices. This last feature is verified comparing the non-cooperative outcomes stemming from different original population densities. © 1989.